You are using Newton's method to solve x3-5x-2=0. If your first guess is x1=2, what value will you calculate for the next approximation x2?
your equation would be
xnew = x - (x^3 - 5x - 2)/(3x^2 - 5)
so for x=2
xnew = 2 - (8-10-2)/(12-5)
= 2.5714
Incidentally, my next 4 answers were
2.4268029
2.4143045
2.4142136 and
2.4142136 which would be one of the answers correct up to 7 decimal places.
90+89=
To calculate the next approximation, x2, using Newton's method, we need to use the formula:
x2 = x1 - f(x1)/f'(x1)
where f(x) is the given function, and f'(x) is the derivative of the function.
In this case, the given equation is x^3 - 5x - 2 = 0. So, let's find the derivative of this function:
f'(x) = d/dx (x^3 - 5x - 2)
= 3x^2 - 5
Now, we can substitute the values into the formula. Given that x1 = 2, we can calculate f(x1) and f'(x1) as follows:
f(x1) = x1^3 - 5x1 - 2
= 2^3 - 5(2) - 2
= 8 - 10 - 2
= -4
f'(x1) = 3x1^2 - 5
= 3(2)^2 - 5
= 3(4) - 5
= 12 - 5
= 7
Now, substituting these values into the formula, we can calculate x2:
x2 = x1 - f(x1)/f'(x1)
= 2 - (-4)/7
= 2 + 4/7
= 14/7 + 4/7
= 18/7
Therefore, the next approximation, x2, is equal to 18/7.